Effects of shape and solute-solvent compatibility on the efficacy of chirality transfer: Nanoshapes in nematics

Chirality, as a concept, is well understood at most length scales. However, quantitative models predicting the efficacy of the transmission of chirality across length scales are lacking. We propose here a modus operandi for a chiral nanoshape solute in an achiral nematic liquid crystal host showing that that chirality transfer may be understood by unusually simple geometric considerations. This mechanism is based on the product of a pseudoscalar chirality indicator and of a geometric shape compatibility factor based on the two-dimensional isoperimetric quotients for each nanoshape solute. The model is tested on an experimental set of precisely engineered gold nanoshapes. These libraries of calculated and in-parallel acquired experimental data among related nanoshapes pave the way for predictive calculations of chirality transfer in nanoscale, macromolecular, and biological systems, from designing chiral discriminators and enantioselective catalysts to developing chiral metamaterials and understanding nature’s innate ability to transfer homochirality across length scales.

keeping the mixture in low speed to avoid foam formation. It is essential to set the stirring speed to 1,000 rpm when NaBH4 is added. NaBH4 needs to be fresh prepared, and it should be cold when we added to the reaction (0 °C).

4|
Place 50 mL Erlenmeyer flask on the hotplate with the temperature set to 25 ℃, add 1.8 mL of 0.1M CTAC to Erlenmeyer flask then 8 mL of DI H2O to dilute the surfactant concentration.

6|
Place a 100 mL Erlenmeyer flask on a hotplate with the temperature set to 25 ℃, add 40 mL of 0.05M CTAC solution (prepared in step 2 and diluted to half the concentration of CTAC) to the flask.

7|
After 5 minutes the CTAC solution was uniform, and then 500 µL of 0.05M HAuCl4 ⋅ 3H2O solution was added followed by 300 µL of a 0.01M NaI solution.

8|
Before proceeding to the main reaction, the papered seed solution was diluted 10 times in a 0.1M CTAC solution.
9 | 0.1M aqueous solution of ascorbic acid was prepared (176 mg in 10 mL DI H2O) and added subsequently in portions of 40 µL and 400 µL to first growth (steps 4 & 5) and second growth solution (steps 6 & 7) respectively; both mixtures were mildly shaken with a concomitant color change of the transparent solutions from light brown to yellow, indicating the reduction of Au III to Au I .
10| With a micropipette, 200 µL of diluted seed solution (see step 8) was added to the first growth solution and immediately 3.2 mL of this solution was injected in the second growth solution with a 5 mL plastic syringe.
▲CRITICAL STEP: We recommend not to use a magnetic stir bar for the growth solutions. It is essential to be fast (less than 3 seconds) when adding the first growth solution to the second growth solution. If the color of the first growth solution changes to the dark pink (only very light pink is suitable), the reaction has already failed, and it is better to prepare the new growth solution and redo steps 4, 5, 9, and 10.
Purification and storage of gold nanoprism solution • timing 20 h 11| After 5 h, when the reaction is complete, 11 mL of the CTAC stock solution was added and the resulting solution gently shaken. Then, this solution was transferred to a 100 mL glass cylinder flask for 15 h, and then the precipitation was collected (greenish blue color). The gold nanoprisms stuck to the wall of glass cylinder and were washed using 10 mL DI H2O.
12| Mild agitation by shaking produced a colloidal aqueous solution of the gold nanoprisms, which was transferred to 2 mL volume centrifuge tubes. Centrifugation at 12,000 rpm for 10 minutes was used to isolate the nanoprisms.
13| The supernatant was removed and discarded by disposable pipettes and any remaining product was washed twice and redispersed with DI H2O.
◆PAUSE POINT: The gold nanoprisms are stable under ambient conditions (room temperature) for several months without changing the shape or quality.

1|
The sand bath was set to a temperature of 40 ℃, then a larger magnetic stir bar was placed into a 100 mL Erlenmeyer in the sand bath and allowed to temperature equilibrate for 15 min.

2|
Once the temperature was reached and constant, 8.4 g of Triton X-100 was added using a 10 mL pipette. The spin rate of stir bar set between 500 -600 rpm and the heat of sand bath was set to 45 ℃.

3|
A 1.25mM gold salt solution was prepared in the 20 mL disposable glass vial (9 mg of HAuCl4 ⋅ 3H2O dissolved in 11.6 mL of DI water). The gold solution was injected to the surfactant solution of Triton X-100 in water in 4 steps over 40 seconds. The spin rate of the stir bar was increased to 900 rpm and the mixture stirred for 1 h at 35 ℃ to obtain a homogeneous mixture.

4|
A 2mM silver salt solution was prepared in a 20 mL disposable glass vial by dissolving 3.4 mg of AgNO3 in 10 mL of DI water. This solution was added to the mixture (step 3) and the spin rate was kept at 1,000 rpm for 5 minutes to produce a homogenous mixture.

5|
In a 10 mL disposable glass vial a 20mM solution of L(+)-ascorbic acid in DI water was prepared and 2.5 mL of this solution was added to the mixture rapidly.

6|
The mixture (step 5) was removed from the sand bath and shaken mildly, and it was kept at 25 ℃ for 5 h undisturbed.

7|
The colloidal mixture was diluted 2-times with DI H2O then was transferred to 2 mL volume centrifuge tubes to concentrate the product via centrifugation at 12,000 rpm for 15 minutes. Then, the supernatant was removed and discarded by disposable pipettes. Any remaining product was washed 3-times and redispersed with DI H2O.
▲CRITICAL STEP: The temperature should be carefully set up to avoid aggregation of gold solution in the hexagonal columnar phase of the liquid crystal template (Triton X-100). Also, if Triton X-100 is not warm enough it causes a gradient concentration of the surfactant. Triton X-100 is very viscous, so it is recommended that one uses a sufficiently large amount of Triton X-100 for the transfer to the reaction flask. In preparation steps 1 to 4, adding AgNO 3 does not produce any obvious color change, but by adding ascorbic acid the mixture progressively turned from yellow to transparent; at first pink, and finally to a greenish blue color that darkens with time.
◆PAUSE POINT: Gold nanostars are stable under ambient conditions (room temperature) for several month without changing the shape or quality.
✦COLOR INDEX FOR SOLUTIONS (Step: S):
2| The temperature of the sand bath was set to 30 ℃, then a sufficiently large magnetic stir bar was placed inside a 100 mL Erlenmeyer flask. Once the temperature equilibrated (ca. 15 min), 50 mL of the gold nanoprism solution were added.
3| Thereafter, 500 µL of 1M HCl and 400 µL of H2O2 (10 wt.% in water) were prepared and injected into the nanoprism solution. The spin rate of stir bar was set between 500 and 600 rpm and the mixture stirred for 48 h.

4|
The mixture was then transferred to 2 mL volume centrifuge tubes and concentrated by centrifugation at 10,000 rpm for 10 minutes. Then, the supernatant was removed and discarded using disposable glass pipettes. The remaining product was washed 2 times and redispersed in DI H2O.
▲CRITICAL STEP: Gold nanodisks with varying diameters (undesired large diameter distribution). By carefully controlling the oxidation reaction every 12 h (using TEM) and using exact concentrations of the HCl and H2O2 solutions (step 3) gold nanodisks with quasi-identical diameter and thickness can be obtained.
◆PAUSE POINT: Gold nanodisks with smaller diameter are stable under ambient conditions for a week without changing shape or quality when stored at room temperature and at neutral pH.

1|
Place a 20 mL disposable glass vial with a magnetic stir bar on a hot plate stirrer and set the stir bar speed to 300 rpm; set the temperature of the sand bath at 30 ℃.

3|
Prepare 100 mL of 0.1M CTAB solution in DI H2O. Once the temperature of the sand bath and flask has equilibrated (ca. 15 min), 9.5 mL of the CTAB solution was injected to the vial and mixed with gold solution.

4|
After 5 minutes, 500 µL of a freshly prepared, ice-cold 0.01M NaBH4 solution (7.6 mg in 10 mL of 0.01M NaOH in DI water solution) was prepared and injected under vigorous stirring at 1,100 rpm. This seed solution was kept for 2 h at 25 ℃ to consume excess borohydride.

Preparation of growth solution • timing 30 min
5| In disposable glass vials, stock solutions of 0.1M AgNO3 (17 mg in 10 mL DI H2O) and 0.1M ascorbic acid (176 mg in 10 mL DI H2O) were prepared.

6|
Five 20 mL precleaned disposable glass vials were used to prepare growth solution: to each vial 8 mL of CTAB, 0.5 mL of HAuCl4 ⋅ 3H2O and 20 µL of AgNO3 were added, respectively, and the sealed vials (solutions) were gently inverted to homogenize the solutions.

7|
To each solution, 50 µL of 1M aqueous HCl and 80 µL of the ascorbic acid solution were introduced, sealed, and gently inverted again.

8|
Once these solutions become transparent, 2 mL of seed solution (Steps 1 -4) was added and vigorously mixed for 30 second.

9|
The solutions were kept undisturbed for 16 h at exactly 28 ℃, and the nanorods were isolated the next day by centrifugation at 11,000 rpm for 10 minutes. The resulting pellets in the centrifuge tubes were collected and redispersed in vials in DI water using 2 mL volume plastic vials. The colorful supernatant was collected and purified in the same way 2 times to collect as much of the nanorods as possible.
▲CRITICAL STEP: Smaller gold nanorods with varying aspect ratios might be obtained if the nanoparticle seeds in the seed solution are not uniform (size and shape -check by TEM). The HCl concentration of the growth solution remains critical since the reduction potential is pH dependent. By controlling the centrifuge speed, time, and concentration of the HCl solution, gold nanorods with nearly identical aspect ratio can be produced.

1|
Use steps of 1 to 4 as described above in section 2.5 for the seed solution preparation.
▲CRITICAL STEP: Tuning the pH of the DI H2O using 1M HCl while the CTAB solution is prepared (pH = 5.5).

3|
Five 20 mL disposable glass vials were precleaned for the preparation of the growth solution; to each vial, 8 mL of 0.1M CTAB, 0.5 mL of 0.01M HAuCl4 ⋅ 3H2O and 100 µL of 0.1M AgNO3 (all aqueous) were added, respectively, and the sealed vials (solutions) were gently inverted.
4| 40 µL of 1M HCl was introduced to each vial, and the sealed vials again gently inverted.
These solutions then remained at rest for 30 min.

5|
Thereafter, 500 µl of hydroquinone was added to each vial, and the vials gently swirled.

6|
Once these solutions become transparent, 2 mL of seed solution was injected to the mixture. The solutions were kept undisturbed for 16 h at exactly 28 ℃ and were purified the next day. To do so, the mixture was transferred to 2 mL plastic centrifuge vials and centrifuged at 10,000 rpm for 10 minutes. The colorful supernatant was collected and purified using the same method 2 times to collect as much of the nanorods as possible.
▲CRITICAL STEP: The pH of DI H2O was adjusted to 5.5 for the CTAB solution. To obtain nanorods with high aspect ratio, the weak reducing agent hydroquinone was used instead of ascorbic acid. The exact concentration of the HCl solution is critical for producing nanorods with higher aspect ratio.
◆PAUSE POINT: Under ambient conditions these nanorods are stable for 3 months without changes the shape (aspect ratio) or quality.

S3.1.2 Size distributions for the various gold nanoshapes
The average size distributions were obtained by ImageJ ® (66); table colors match the absorption band of the nanoshape in the visible portion of the EM spectrum.
Tables S1 -S6. Size and size distribution of the cholesterol-thiol-capped gold nanoshapes. Size distributions derived from the analysis of the TEM images using the image analysis software ImageJ ® . Average aspect ratio, AR, shown below is for the gold nanoshape cores. 178.0 ± 12.8 The   Fig. S3. UV-vis-NIR spectra of the precursor, surfactant-capped nanoshapes in H2O and for all chiral cholesterol-capped particles: a CTAB-capped (blue spectrum) and cholesterolthiol-capped LAR-GNR (red spectrum), b CTAB-capped (blue spectrum) and cholesterol-thiolcapped HAR-GNR (red spectrum), c CTAB/CTAC-capped (blue spectrum) and cholesterol-thiolcapped GNPR (red spectrum), d CTAB/CTAC-capped (blue spectrum) and cholesterol-thiolcapped HAR-GND (red spectrum), e CTAB/CTAC-capped (blue spectrum) and cholesterol-thiolcapped LAR-GND (red spectrum), and f Triton X-100-capped (blue spectrum) and cholesterolthiol-capped GNS (red spectrum). Ligand exchange is indicated by transfer of the nanoshapes to the organic phase as well as shifts of the various SPR bands; hypsochromic for GNRs and GNPR and bathochromic for GNDs and GNS.   a The calculated wt.%Ligand range is taking the size distribution of the nanoshapes (obtained by TEM image analysis) and therefore the calculated surface area ranges into account (see equations below). b The ligand coverage of the synthesized and purified nanoshapes was obtained by thermogravimetric analysis (TGA); pure gold was obtained as a residue after heating above 500 °C.  Fig. S6. Pitch measurements -free surface. Polarized optical photomicrographs of the induced N*-LC textures used for helical pitch measurements using the free surface method (crossed polarizers): a -f 5CB doped with 0.5 wt.% of the different nanoshapes (scale bars = 50 μm).

Fig. S7. Pitch measurements -free surface.
Polarized optical photomicrographs obtained on cooling from the isotropic liquid phase at 20 °C (crossed polarizer P and analyzer A) for helical pitch measurements using the free surface method: a -d 5CB doped with 0.4 wt.% of different nanoshapes (scale bars = 100 µm). Doping less than 0.5 wt.% of the cholesterol-capped nanoshape dopants results in large helical pitch values that cannot be measured with any of the pitch measurement methods described here. Onsets of fingerprint textures can be seen in a and b for LAR-GNR and HAR-GND, but images c and d show textures typical for achiral nematic LC phases (Schlieren textures with disclination lines, i.e., +1 and -1 defects) for HAR-GNR and LAR-GND.  The dotted red and the solid blue lines indicate the logarithmic and steadily changing slope of the data points, respectively (data taken from figs. S9 and S10). The hypothetical dashed grey lines show the common trends of the inverse pitch vs. the concentration of a typical organic molecule-based chiral additive (or dopants), where the p −1 linearly increases with the concentration and then saturates (plateaus) once a certain pitch is reached (i.e., the pitch saturates). This behavior is normally limited to the lower chiral additive concentration regime as higher concentrations frequently lead to either phase separation of the chiral additive or to the induction of other chiral phases (e.g., blue phases; (68)). As for the previously reported GNRs (40), the experimentally observed increasing slope with increasing nanoshape concentration reveals a cooperativity exerted by the chiral ligand-capped gold nanoshapes, now helically distorted within the induced N*-LC matrix that further contributes to the observed amplification of the chirality transfer efficacy.

S7.3.1 2-D Isoperimetric ratios
The values of IPR2D and IPR3D are summarized below in Tables S10 and S11 and the differences to the IPR (2-and 3-D) of 5CB compared against the 5,+ data shown in Fig. 4 (lowest difference set to 100%; highest difference to 0%). Calculations of IPR2D and IPR3D values took the energyminimized length and shape of the cholesterol ligands (forming a monolayer capping each nanoshape) into account. For modified values of the AR, see Table 1 in the main manuscript.  < with respect to 5CB GNP5 and GNP10 ≈13.5 c 9.9 LAR-GNR 14.0 d 9.4 MAR-GNR 24.0 d 0.6 HAR-GNR 36.5 -44.0 d 13.1 -20.6 LAR-GND and HAR-GND 12.6 10.8 GNPR 20.8 2.6 GNS (average 6 spikes) 18.7 e 4.7 a Assuming an ellipsoidal shape with an aspect ratio of AR ~ 4 (l = 1.8 nm, weff = 0.45 nm (70)) calculated for 5CB (energy-minimized) conformation in the nematic phase with free rotation about the long molecular axis. b Assuming an ellipsoidal shape with aspect ratios ranging from AR ~ 3 to AR ~ 6 calculated for (energy-minimized) conformations in the nematic phase with free rotation about the long axis. c Depending on the polyhedral shape that can include a range of Platonic and Archimedean solids (71), the most energetically favored truncated octahedron was used (72). d Based on the average dimensions (length and diameter) of the GNR. e Reflects experimental evidence that the number of thorns is on average 6 -8, but the average number of projected spikes for the 2D projection is about six. Furthermore, their individual length varies, which cannot be precisely included in the calculation. Thus, the average length was used.